Re: Exception to the rule? (Tarski´s T-scheme)

From: Jeffrey Ketland (ketland_at_ketland.fsnet.co.uk)
Date: 06/26/04 

Date: Sat, 26 Jun 2004 01:05:46 +0100

Andrew Boucher wrote:

>> Do you agree that
>>
>> (i) "Snow" contains four letters
>> implies
>> (ii) There is something which contains four letters
>
>Yes. But, unless I'm missing something, that doesn't seem to be
>relevant to what I'm saying.

OK. Maybe I'm confused about what you're saying. I took the criticism to
concern whether existential generalization on quotation names or
structural-descriptive terms is legitimate.

Let's consider this criticism first. Presumably (unless we insist on free
logic for names), we have,

    (i) "Snow is white" is true iff snow is white
implies
    (ii) There is something x such that x is true iff snow is white

If that's OK (it's classical logic), then the following is OK:

    (iii) "Snow is white" is true iff snow is white
    (iv) "Snow is not white" is true iff snow is not white
jointly imply
    (v) "Snow is not white" is distinct from "Snow is not white"

Finally, from (v), we have:
    (vi) There are x, y such that x is distinct from y

This is a classically valid argument from two T-sentences, (iii) and (iv).
The only step which is not valid in free logic is the step which you seem to
accept (i.e., existential generalization on quotation names).

So, it seems to me that this first criticism, that the T-scheme doesn't
imply the existence of at least two objects, depends upon rejecting EG on
quotation names.
But as I've remarked, this would be extremely weird.

If we didn't want to use denoting terms in formulating the T-scheme for a
particular language, we could use structural-descriptive names and
predicates for being an "s", and being an "n", and so on.
So, let's go Russellian/Quinian. Suppose that "S(x)" means "x is an "S" ",
and "N(x)" means "x is an "n" ", and so on. Then an instance of Tarski's
T-scheme would be a highly convoluted mess:

   (T)* For some x_1, x_2, ...., x_13, such that S(x), N(x_2), O(x_3),
W(x_4),
        Space(x_5), .... there is a concatenation y, namely
        x_1^(x_2^(....x^_13))...), such that y is true if and only if snow
is white.

And similarly through all the other sentences of the language in question.
As I have said, this, along with another convoluted instances (now for "Snow
is not white", i.e.,

   (T)** For some x_1, x_2, ...., x_17, such that S(x), N(x_2), O(x_3),
        W(x_4), Space(x_5), N(x_6), O(x_7), .... there is a concatenation
        y*, namely x_1^(x_2^(....x^_17))...), such that y* is true if and
        only if snow is not white.

Then (T)* and (T)** imply that y and y* are distinct (and explicitly assert
the existence of both y and y*).

There is also a second criticism, which might be your point. This is based
on a different response: namely, that just which set of sentences count as
the relevant instances of the T-scheme depends upon the object language.
This is right, of course. If we consider a language L with just one sentence
A (so that we don't even have ~A), there is just one T-sentence, namely,

        Tr(t) iff A

where Tr, t, and "iff" are expressions in the meta-language. If we consider
a language L with just one sentence, then I agree that the relevant T-scheme
for that language doesn't imply the existence of two objects.

But if our object language has just A and ~A, then we shall also have terms
denoting them, say t1 and t2, and the T-sentences are

      Tr(t1) iff A
      Tr(t2) iff ~A

Then, we can prove t1 =/= t2, and thus that there exist two objects (modulo
free logic).

At the start of this thread, I was simply assuming a language with the usual
properties. I.e., a non-trivial language, which necessarily has denumerably
many sentences. The T-scheme doesn't imply the distinctness of all of these,
but it does imply that there are at least two of them.

I've discussed what I take to be the two criticisms:
(a) We are applying EG to quotation names. I replied that not doing this
would be weird.
(b) We might consider a language L with just one sentence (or, e.g., without
a negation connective). Then there is just one T-sentence, and it doesn't
imply the existence of two distinct objects. Agreed. But modulo (a), it does
imply the existence of at least one: namely, the sentence in question.

--- Jeff

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